On 05/11/2024 06:44 PM, Ross Finlayson wrote:
On 05/11/2024 06:17 PM, Ross Finlayson wrote:
On 05/11/2024 04:47 PM, Jim Burns wrote:
On 5/11/2024 7:11 PM, Ross Finlayson wrote:
On 05/11/2024 02:44 PM, Ross Finlayson wrote:
On 05/11/2024 02:05 PM, Ross Finlayson wrote:
On 05/11/2024 12:24 PM, Jim Burns wrote:
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The case is that induction goes through,
an inviolable law you call it:
does it go all the way through?
Does it complete?
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It is complete.
There is no completing.activity,
so I wouldn't say it completes.
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Compare to right triangles:
Are all the squares of two shorter sides
summed to the square of the longest side?
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That's a tricky question to answer because
there is no summing done.
That relationship between the sides
is simply something true about right triangles.
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And it is complete == it is true for each.
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We don't typically ask the tricky question
about right triangles.
We ask the tricky question about cisfinite induction
because we imagine it as a process,
which we don't for right triangles.
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Cisfinite induction is NOT a process.
Cisfinite induction is an argument,
completely correct or completely incorrect.
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See, the contrary inductive analyst just says
"in case you don't have a deductive argument why
something is so, induction is so much shifting-sands
and slippery-slope." He just has "the base case is
you haven't completed induction, and so is the
subsequent case, case closed: case not closed".
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When the argument is completed,
induction is completed.
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There is something completely different
which is also called induction.
The completely.different induction is physics.
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Physics.induction is not unbreakable.
Physics.induction isn't cisfinite or transfinite induction.
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What I recall of the context of the Pythagorean theorem,
was that after algebra already was trigonometry, and
the definitions of the trigonometric functions, for
sine and cosine and tangent, about the opposite and
adjacent and hypotenuse, then as of a right triangle
with its hypotenuse the radius of a unit circle, that
the right angle is as with regards to the abscissa
and ordinates or where the lines drop or slide to
the x or y axis of the usual X-Y coordinate setting
of a circle centered at the origin, it was of the
secondary school's first three years of geometry,
algebra, and trigonometry, or along those lines.
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So, we computed a bunch of ready things about
those often with the Pythagorean theorem,
which is as an addition-formula, mostly about
30-60-90 triangles, and, isosceles triangles,
or 45-45-90, then those got used throughout
precalculus and a couple years of calculus
or high school.
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So anyways one time I see a diagram about
Pythagorean triples, those being tuples of
three integers that have a^2 + b^2 = c^2,
and what they'd done was right triangle,
then draw a square as of the square alongside
it, and counting the boxes of the squares of
a b c it's that the boxes of the squares of a
and b equals the boxes of the square of c.
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If that's not a proof of the Pythagorean theorem
and least it's graphically intuitive for some values,
where of course there are hundreds of known
proofs of the Pythagorean theorem, since the
time of Pythagoras as some even have as from
greater antiquity, then it reminds of things
like Rodriguez formula, Vieta's formulas,
Nicomachus' theorem and formulas,
Pascal triangle and bonomial theorem,
all what are sorts of addition formulas,
like an addition formula of the product
of exponents as the sum of the powers.
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So, that Pythagorean triples exist, and it results
that the rightness of a triangle with sides length
the Pythagorean triple can be established without
invoking the Pythagorean theorem, doesn't so
much make it so the other way around, from
induction over Pythagorean triples, without
showing as how all right triangles are somehow
as some congruence to what is some Pythagorean
triple, of the equivalence class of all the triples
and all the congruences to triangles with a
unit length longest side, establishing infinite
expressions, and closures, of completion,
to make a case for the Pythagorean theorem
as via induction from an explication after
the enumeration of Pythagorean triples,
which via inspection have a^2+b^2 = c^2,
as for that it results congruences that
"go to" any given dimensions of a right
triangle.
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About the cisfinite and transfinite induction,
and I know it's not the languages fault that
there's the associated psychosexual connotation,
I'm glad you make the point though that
it just is what it is, and, a case for induction
more or less needs some reason its tendency,
to succeed as it were, then that induction
is given its course, then that the course-of--passage,
of what the plain old infinite induction, arrives.
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I.e., it's always "infinite induction", after cause-and-effect,
with that also being induction or a case, mathematical
induction, and there can't be any reasonable counterclaims
or they'd be just as guaranteed as the contradistinct opposite.
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So, it makes for a very strong perceived requirement
for deductive reasoning _why_ convergence criteria
exist, besides that "given an infinite expression,
it's an infinite expression".
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Here then that's most Zeno's about geometric series,
and then about things like Stirling numbers and of
course the discussions we've been having over the
past few months about the convergence and
the slooowwwly convergent and all this,
the "scaffolding" of the infinite expressions
we've been discussing and at length.
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Warm regards
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Now that you mentioned it, the right triangle you
figure usually is the result of classical constructions,
making a line and a perpendicular line and measuring
any length along one line and any length along another
and connecting those to make a right triangle.
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Yet, if you can make the trigonometric functions,
then they also define right angles, according to
that the inverse of one or the other is pi/2.
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So, you usually figure the trigonometric functions,
which are continuous functions, are drawn basically
as of about the triangle with hypotenuse a radius
and the other point an abscissa or ordinate.
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Yet, there's another way to draw the trigonometric
functions or sine and cosine curves, that isn't via
classical constructions. It's a sort of "tri-(equi)lateral-ometry",
or as I called it also "N-gonometry", and it's accomplished
in this manner: with an equi-lateral triangle, unhinge one
corner then un-roll it, one point fixed, while opening
each of the angles at the same rate until it results a
segment length 3, meeting a line, then flip that forward
one and then roll it back up, and the traced point is
the sine curve or its complement the cosine curve.
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So, "right triangles", could also be defined this way,
though it's a sort of round-about manner and involves
more than our classical tools of compass and edge,
is built from equilateral triangles, and trigonometry.
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You know, just saying, ....
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Then, the other idea was "what about the very narrow
and very tall triangle, so tall that both the base angles
look right". Now, one or the other of the legs is longer,
though as it's very tall their difference is small, which
gets into: "x^2 +- 1". (Vis-a-vis the parallel postulate.)
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Anyway these reflect how for example "classical geometry
is its own theory" then for something like "Zeno's arguments
make their own theory" and "an extension of Zeno's argument
where the geometric series is well-defined and complete and
has a sum that makes for a destination for inductive inference
is its own theory".
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This idea about Pythagorean triples and trigonometry and
an analytic framework for trigonometry and Pythagorean
theorem, gets into triangles and their angles and sides,
as what's usually developed all from classical constructions,
and what are the extensions to classical constructions
like the Archimedean spiral for example and here a notion
of un-rolling-and-re-rolling un-hinged equi-laterals, making
a usual sort of periodic symmetrically-reflecting inchworm-hump,
that here it's about classical constructions, and, analysis of
planar figures.
So, triangles, classically constructed, will usually have one
or two or three sides of rational length, then where it's so
that the sides are rational, then multiplying the side lengths
by the least-common-multiple of the denominators and
then dividing the side lengths by the greatest-common-divisor,
of the numerators, results a triagonal whose side lengths
are a Pythagorean triple, integers.
Then, those triangles with irrational sides, it sort of works
up analyticity what it's called, the completion of those,
with regards to those being "the limit" of the rational side
lengths, to, irrational side lengths.
So, it's figured that side lengths that are Pythagoren triples,
always have a^2 + b^2 = c^2, then as to establish that triangles
with side lengths a Pythagorean triple, are always right.
https://en.wikipedia.org/wiki/Pythagorean_tripleThen it sort of work out also that multiples of those
make for analyticity of completions of Heronian triples,
where the area, besides the side lengths, is also an integer.
(Of course that's already a given, yet, it's given.)
Of course these result arbitrarily "large" numbers.
So, the point is that there are other sources of the
trigonometric functions, than the usual development
as after right triangles and the Pythagorean theorem,
then as with regards to what constructions, augment
the classical constructions, to so provide, said functions.
https://en.wikipedia.org/wiki/Trigonometric_functionsSo, this "tri-(equi)lateral-ometry", also sort of extends
to the the n-gon from the tri-gon, "n-equi-lateral-ometry",
and besides regular polygons the squashed which would
also have the property of un-rolling-re-rolling down the line,
making all sorts pairs of these functions like sine and cosine.
Sine and cosine or course are very well explored, and also,
they're orthogonal functions, so the integral of their
product is various 0 or 1 when augmented with small-angle
approximation the linearisation that's the most usual sort
of establishment of the Fejer and Dirichlet kernels for the
Fourier-style analysis and many other aspects of the
convolutional setting and transform and analysis and
what's called functional analysis, so it may be of interest
to have parametric recursive forms for all sorts these
as stemming from otherwise classical geometric and
classical constructions, an extra construction, and,
the relation of the resulting scales of the amplitudes
to the exact trace of the "right triangle" trigonometry,
the, "equilateral triangle", trigonometry.
In this way one can find a sub-field of mathematics, briefly.
Its own theory, ....
And Zeno's just like "hey, I do back-and-forth, and recursively,
and then can get into the straight or round track, triangles and
the planar are your own business".
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